Transcendental inverse eigenvalue problems in damage parameter estimation

In this paper the author has introduced transcendental eigenvalue problems for estimating the damage parameters in the continuous structure from measured eigenvalues or natural frequencies. For simplicity axially vibrating rods are considered in which single or multiple damage parameters due to open cracks or notches are simulated by linear springs that are representative of the loss in stiffness or axial rigidity at the location of cracks. Transcendental eigenvalue problems (direct and inverse) associated with rod having multiple damage parameters have been formulated and solved. The numerical method for solving such eigenvalue problems are developed which overcomes the requirements of closed form characteristic frequency equations that are often unavailable. The modeling and solution approach developed here is utilized for evaluating the spectrum of rods with multiple damage parameters as well as for identifying the locations and severity of the damage parameters purely from the eigenvalues. Numerical examples and simulations corresponding to various damage configurations are presented and verified against experimental evidence. It is demonstrated that the solution of transcendental inverse eigenvalue problems can be successfully used for estimating the damage parameters by using only few and selected eigenvalues corresponding to the measured resonant and anti-resonant frequencies of the rod.